FS0007 · theorem body

four_square_additive_gap_reorder

Alpha v34 checked-use · independently kernel and Lean verified; not Stable

The five additive square-gap contributions reorder without subtraction or a polynomial normalizer.

historical independently replay-verified empty-context experiment; the experiment itself persisted no certificate and granted no release authority; current checked use follows separately sealed proof bundles; no Stable promotion.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Statement with defined notation

forall a b c d. ((a + b) + (c + d)) + a = d + ((a + c) + (a + b))

Every purple notation token opens its conservative definition. This reading surface never changes the unchanged intuitionistic kernel or confers checked-use authority.

Definitions used by this theorem

In the theorem statement

none

In local proof propositions

none
Exact expanded first-order statement
forall a b c d. ((a + b) + (c + d)) + a = d + ((a + c) + (a + b))

Proof neighborhood

Direct theorem prerequisites

add_shuffle_middle · Alpha closed add_assoc · Stable closed add_comm · Stable closed FS0006 four_square_add_swap_right_tail

Direct theorem dependents

FS0009 four_square_gap_balance_right

Definition-aware tactic body

Only propositions whose conservative expansion has been checked for exact first-order equivalence are compacted. Every changed line retains its immutable exact replay command.

Read the argument

Proof checkpoints

25 script commands · 11 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (1)
four square add swap right tail
01Fix variables and assumptionsL1–4

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro a
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro d
02Calculate and transport equalitiesL5–6

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L5
    trans ((a + c) + (b + d)) + a
  2. L6
    congr
03Use earlier factsL7–7

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L7
    apply add_shuffle_middle
04Calculate and transport equalitiesL8–9

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L8
    refl
  2. L9
    trans (a + c) + ((b + d) + a)
05Use earlier factsL10–10

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L10
    apply add_assoc
06Calculate and transport equalitiesL11–13

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L11
    trans (a + c) + (b + (d + a))
  2. L12
    congr
  3. L13
    refl
07Use earlier factsL14–14

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L14
    apply add_assoc
08Calculate and transport equalitiesL15–17

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L15
    trans (a + c) + (d + (b + a))
  2. L16
    congr
  3. L17
    refl
09Use earlier factsL18–18

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L18
    apply four_square_add_swap_right_tail
10Calculate and transport equalitiesL19–23

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L19
    trans (a + c) + (d + (a + b))
  2. L20
    congr
  3. L21
    refl
  4. L22
    congr
  5. L23
    refl
11Use earlier factsL24–25

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L24
    apply add_comm
  2. L25
    apply four_square_add_swap_right_tail

Library-wide reading audit

Original defined command ledger · 25 lines
  1. 0001intro a
  2. 0002intro b
  3. 0003intro c
  4. 0004intro d
  5. 0005trans ((a + c) + (b + d)) + a
  6. 0006congr
  7. 0007apply add_shuffle_middle
  8. 0008refl
  9. 0009trans (a + c) + ((b + d) + a)
  10. 0010apply add_assoc
  11. 0011trans (a + c) + (b + (d + a))
  12. 0012congr
  13. 0013refl
  14. 0014apply add_assoc
  15. 0015trans (a + c) + (d + (b + a))
  16. 0016congr
  17. 0017refl
  18. 0018apply four_square_add_swap_right_tail
  19. 0019trans (a + c) + (d + (a + b))
  20. 0020congr
  21. 0021refl
  22. 0022congr
  23. 0023refl
  24. 0024apply add_comm
  25. 0025apply four_square_add_swap_right_tail